Decision-Making with Belief Functions: a Review

Approaches to decision-making under uncertainty in the belief function framework are reviewed. Most methods are shown to blend criteria for decision under ignorance with the maximum expected utility principle of Bayesian decision theory. A distinction is made between methods that construct a complete preference relation among acts, and those that allow incomparability of some acts due to lack of information. Methods developed in the imprecise probability framework are applicable in the Dempster-Shafer context and are also reviewed. Shafer's constructive decision theory, which substitutes the notion of goal for that of utility, is described and contrasted with other approaches. The paper ends by pointing out the need to carry out deeper investigation of fundamental issues related to decision-making with belief functions and to assess the descriptive, normative and prescriptive values of the different approaches

Topics in inference and decision-making with partial knowledge

Two essential elements needed in the process of inference and decision-making are prior probabilities and likelihood functions. When both of these components are known accurately and precisely, the Bayesian approach provides a consistent and coherent solution to the problems of inference and decision-making. In many situations, however, either one or both of the above components may not be known, or at least may not be known precisely. This problem of partial knowledge about prior probabilities and likelihood functions is addressed. There are at least two ways to cope with this lack of precise knowledge: robust methods, and interval-valued methods. First, ways of modeling imprecision and indeterminacies in prior probabilities and likelihood functions are examined; then how imprecision in the above components carries over to the posterior probabilities is examined. Finally, the problem of decision making with imprecise posterior probabilities and the consequences of such actions are addressed. Application areas where the above problems may occur are in statistical pattern recognition problems, for example, the problem of classification of high-dimensional multispectral remote sensing image data

Dynamic bid–ask pricing under Dempster-Shafer uncertainty

We deal with the problem of pricing in a multi-period binomial market model, allowing for frictions in the form of bid–ask spreads. We introduce and characterize time-homogeneous Markov multiplicative binomial processes under Dempster-Shafer uncertainty together with the induced conditional Choquet expectation operator. Given a market formed by a frictionless risk-free bond and a non-dividend paying stock with frictions, we prove the existence of an equivalent one-step Choquet martingale belief function. We then propose a dynamic Choquet pricing rule with bid–ask spreads showing that the discounted lower price process of a European derivative contract on the stock is a Choquet super-martingale. We finally provide a normative justification in terms of a dynamic generalized no-arbitrage condition relying on the notion of partially resolving uncertainty due to Jaffray

A method of classification for multisource data in remote sensing based on interval-valued probabilities

An axiomatic approach to intervalued (IV) probabilities is presented, where the IV probability is defined by a pair of set-theoretic functions which satisfy some pre-specified axioms. On the basis of this approach representation of statistical evidence and combination of multiple bodies of evidence are emphasized. Although IV probabilities provide an innovative means for the representation and combination of evidential information, they make the decision process rather complicated. It entails more intelligent strategies for making decisions. The development of decision rules over IV probabilities is discussed from the viewpoint of statistical pattern recognition. The proposed method, so called evidential reasoning method, is applied to the ground-cover classification of a multisource data set consisting of Multispectral Scanner (MSS) data, Synthetic Aperture Radar (SAR) data, and digital terrain data such as elevation, slope, and aspect. By treating the data sources separately, the method is able to capture both parametric and nonparametric information and to combine them. Then the method is applied to two separate cases of classifying multiband data obtained by a single sensor. In each case a set of multiple sources is obtained by dividing the dimensionally huge data into smaller and more manageable pieces based on the global statistical correlation information. By a divide-and-combine process, the method is able to utilize more features than the conventional maximum likelihood method

Discrete time models for bid-ask pricing under Dempster-Shafer uncertainty

As is well-known, real financial markets depart from simplifying hypotheses of classical no-arbitrage pricing theory. In particular, they show the presence of frictions in the form of bid-ask spread. For this reason, the aim of the thesis is to provide a model able to manage these situations, relying on a non-linear pricing rule defined as (discounted) Choquet integral with respect to a belief function. Under the partially resolving uncertainty principle, we generalize the first fundamental theorem of asset pricing in the context of belief functions. Furthermore, we show that a generalized arbitrage-free lower pricing rule can be characterized as a (discounted) Choquet expectation with respect to an equivalent inner approximating (one-step) Choquet martingale belief function. Then, we generalize the Choquet pricing rule dinamically: we characterize a reference belief function such that a multiplicative binomial process satisfies a suitable version of time-homogeneity and Markov properties and we derive the induced conditional Choquet expectation operator. In a multi-period market with a risky asset admitting bid-ask spread, we assume that its lower price process is modeled by the proposed time-homogeneous Markov multiplicative binomial process. Here, we generalize the theorem of change of measure, proving the existence of an equivalent one-step Choquet martingale belief function. Then, we prove that the (discounted) lower price process of a European derivative is a one-step Choquet martingale and a k-step Choquet super-martingale, for k ≥ 2